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CMAJ
Modelling an influenza pandemic: A guide for the perplexed
Pandemic Influenza Outbreak Research Modelling Team (Pan-InfORM)
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See other H1N1 articles: Editorial, page 123; Research, page 159
A
new swine-origin influenza A (H1N1) virus, initially identified in Mexico, has now caused outbreaks of disease in at least 74 countries, with declaration of a global influenza pandemic by the World Health
Organization on June 11, 2009.1 Optimizing public health
responses to this new pathogen requires difficult decisions
over short timelines. Complicating matters is the unpredictability of influenza pandemics: planners cannot base
their decisions solely on pre-pandemic factors or on experience from earlier pandemics. We suggest that mathematical
modelling can inform and optimize health policy decisions
in this situation.
Uses of models
Mathematical models of infectious diseases are useful tools
for synthesizing the best available data on a new pathogen,
comparing control strategies and identifying important areas
of uncertainty that may be prioritized for urgent research.
Generation
0
1
2
DOI:10.1503/cmaj.090885
Initial phase of epidemic (R0 = 3)
Disease is endemic (R = 1)
Figure 1: The number of new infections generated when the
basic reproductive number (the number of new cases created
by a single primary case in a susceptible population) is 3. Cases
of disease are represented as dark circles, and immune individuals are represented as open circles. When there is no immunity in the population (left) and the basic reproductive number
(R0) is 3, the initial infectious case generates on average 3 secondary cases, each of which in turn generates 3 additional
cases of disease. Once the disease becomes endemic owing to
acquired immunity (right), each case generates on average
1 additional case (effective reproductive number [R] = 1).
Key points
•
When a new infectious disease emerges, mathematical
models can project plausible scenarios, guide control
strategies and identify important areas for urgent research.
•
Models of influenza pandemics suggest roles for antiviral
drugs and vaccines. Models also raise concerns about
antiviral resistance.
•
Knowledge translation is a key part of modelling activities
that aim to optimize policy decisions for containment of
new infectious diseases.
As an example of synthesizing data, consider the process
of developing a mathematical model of the effectiveness of
influenza vaccines: modellers must draw together information
on influenza epidemiology (including patterns of spread in
different age groups), the natural history of influenza, the
effectiveness of vaccines in randomized trials and the duration of immunity following vaccination or natural infection,2,3
which cannot all be derived from a single study. Once the
model is developed, rapid and inexpensive “experiments” can
be performed by simulating alternative vaccination strategies
(e.g., vaccinating children most likely to transmit influenza,
or vaccinating older adults most likely to have severe complications of influenza).2
The uncertainty involved in this process can be assessed
through sensitivity analysis (in this case, by varying estimates
of vaccine effectiveness across plausible ranges) to examine
whether such variation results in markedly different outcomes. Uncertain model inputs that are extremely influential
in determining the best course of action should be prioritized
for future research.
Elements of models
Elements of epidemic models often include “compartments” or “states” that describe the susceptibility, infectiousness or immunity of individuals in a population, and
“parameters” (numbers) that describe how individuals
move between these states.
A key model parameter is the basic reproductive number, referred to as R0.4 This is the number of new, secondary
infections created by a single primary infectious case introduced into a totally susceptible population (Figure 1). The
importance of R0 relates to the information it provides to
planners: R0 determines the potential of a new pathogen to
CMAJ • AUGUST 4, 2009 • 181(3-4)
© 2009 Canadian Medical Association or its licensors
171
Analysis
The new influenza strain and seasonality
Basic reproductive
number
Mathematical models have provided important insights into
the seasonality of influenza by showing that small seasonal
variations in viral transmissibility can drive large annual
surges in the incidence of the disease.3 The 1918 influenza
pandemic began in the spring, was relatively quiescent over
the summer months and became more severe the following
fall. Models suggest that the new influenza strain could follow a similar pattern in countries in the northern hemisphere
(Figure 3). Such models raise concern regarding the potential
for the new strain to cause serious epidemics in the near term
in the southern hemisphere: South American countries, Australia and New Zealand will enter their upcoming influenza
season without a vaccine against this strain.
1.4
1.2
R0
1.0
0.8
0.6
Proportion
cause an epidemic (if R0 is greater than 1). It can also be
used to estimate the final size of the epidemic with and
without control measures (Figure 2). Models provide a systematic way to estimate R0, which reflects the product of
duration of infectiousness, contact numbers per unit time
and infectiousness per contact. As each of these quantities
increases, R0 increases.
R0 is “unit-less” with respect to time. Therefore, knowing how fast a disease spreads requires knowledge of “generation times.” A generation is the average time from the
onset of infectiousness in a case to the onset of infectiousness in a person that the case has infected. Generations can
be estimated as the sum of the disease’s latent (infected, but
noninfectious) periods and half the duration of infectiousness. Measles and tuberculosis both have similar (very
high) R0 values. However, measles is more infectious per
contact, and tuberculosis has a much longer generation time
because of a longer duration of infectiousness and a prolonged latent period.
Diseases that have low R0 values but short generation
times can have relatively rapid increases in the number of
cases. Published estimates suggest that the R0 for the new
influenza A (H1N1) virus is 1.5 (slightly lower than estimates
for the 1918 influenza pandemic5).6 The generation time is
estimated to be about 2 days,6 although preliminary unpublished estimates in the province of Ontario suggest longer
generation times (Rachel Savage, Ontario Agency for Health
Protection and Promotion, personal communication, May
2009).
Cumulative attack rate
0.4
0.2
0
Prevalence x 10
0
0.5
1.0
Basic reproductive
number
Incidence when R0 = 1.3
Incidence when R0 = 1.6
Winnipeg, fall 1918 (incidence x 3)
10
8
6
4
2.5
3.0
3.5
4.0
1.4
1.2
R0
1.0
0.8
0.6
2
0
0
200
400
600
800
Time, days
Figure 2: The effect of changing the basic reproductive number
(R0) on the severity and duration of an influenza epidemic. A
higher R0 (1.6, red curve) results in an epidemic with a higher
peak incidence and a greater cumulative attack rate (not
shown). When the R0 is lower (1.3, black curve), the epidemic
peaks later, lasts longer and has a lower cumulative attack
rate. The basic reproductive number of the disease is reduced
through control interventions, including isolation of cases and
“social distancing” (e.g., school closures). The blue curve represents actual incidence data from the autumn wave of the 1918
influenza pandemic in Winnipeg (multiplied by 3 for comparability of scales).
Proportion
Incidence per 1000 population
2.0
Time, years
12
172
1.5
Cumulative attack rate
0.4
0.2
0
Prevalence x 10
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Time, years
Figure 3: Possible effects of seasonality on a novel influenza
strain. Panels show simulated epidemic curves when the basic
reproductive number (R0) for influenza oscillates from 1.4 in
winter to just under 1 in summer (black curves). When an epidemic emerges in winter (top panel), it is associated with a
monophasic increase in prevalence (red curve). However, springtime emergence (bottom panel) results in a biphasic epidemic
(red curve), as happened in the 1918 influenza pandemic.
Cumulative attack rates are represented by blue curves. Prevalence data are multiplied by 10 for comparability of scales.
CMAJ • AUGUST 4, 2009 • 181(3-4)
4.0
Analysis
Mitigation without vaccination
Nonpharmaceutical interventions considered in published
models of influenza transmission include isolation of infectious cases, quarantine of potentially infectious but asymptomatic contacts, and “social distancing” measures such as
school closures.7 Models suggest that nonpharmaceutical
interventions that reduce R0 through different, complementary
mechanisms can sometimes be synergistic.
Without a vaccine, antiviral drugs represent an important
tool for controlling influenza pandemics. The new influenza
strain appears to be susceptible to the neuraminidase inhibitors oseltamivir and zanamivir. Models in which resistance is
not considered suggest that the early use of antiviral drugs
could contain influenza epidemics caused by strains with low
R0 values. When resistance is considered in models, it becomes apparent that limiting the initial use of antiviral drugs,
to avoid epidemics caused by antiviral-resistant influenza
strains, is the preferred strategy.8
Modelling and policy
Funding: Pan-InfORM is supported by the Canadian Institutes for Health
Research, the Natural Sciences and Engineering Research Council of Canada
and the Mathematics of Information Technology and Complex Systems
(MITACS) network.
REFERENCES
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3.
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Despite the extensive use of models to evaluate epidemics,
the integration of modelling into policies for disease control
has been limited.9 Participants at a 2008 Canadian pandemic
planning workshop noted that models are most useful when
they are developed cooperatively by modellers and policymakers. The pandemic threat has shown that the ties between
modellers and public health decision-makers are closer than
originally believed: Canadian modellers, including our group,
have helped characterize the epidemiology of this new
influenza strain and have provided input to public health
authorities on control strategies, including the use of antiviral
drugs and school closures.
We hope that models will become more widely available as
desktop decision-support tools for front-line public health and
medical personnel. Knowledge translation is a key part of
modelling activities. Making these models better understood
and more accessible will provide a valuable additional weapon
in the fight against emerging infectious diseases.
Competing interests: None declared.
Contributors: All of the authors were involved in the drafting and revision
of this manuscript and approved the final version submitted for publication.
World now at the start of 2009 influenza pandemic [press release]. Geneva (Switzerland): World Health Organization; 2009 June 11. Available: www.who.int/mediacentre
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Dushoff J, Plotkin JB, Levin SA, et al. Dynamical resonance can account for seasonality of influenza epidemics. Proc Natl Acad Sci U S A 2004;101:16915-6.
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Fraser C, Donnelly CA, Cauchemez S, et al. Pandemic potential of a strain of
Influenza A (H1N1): early findings. Science 2009 May 14. [Epub ahead of print]
Mills CE, Robins JM, Lipsitch M. Transmissibility of 1918 pandemic influenza.
Nature 2004;432:904-6.
Halloran ME, Ferguson NM, Eubank S, et al. Modeling targeted layered containment of an influenza pandemic in the United States. Proc Natl Acad Sci U S A
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Moghadas SM, Bowman CS, Röst G, et al. Population-wide emergence of antiviral
resistance during pandemic influenza. PLoS ONE 2008;3:e1839.
Moghadas S, Pizzi N, Wu J, et al. Managing public health crises: the role of models in pandemic preparedness. Influenza Other Respir Viruses 2009;3:75-9.
Correspondence to: Dr. David Fisman, Ontario Agency for Health
Protection and Promotion, Epidemiology and Surveillance,
480 University Ave., Rm. 406, Toronto ON M5G 1V2;
fax 647 260-7600; [email protected]
Members of the Pandemic Influenza Outbreak Research Modelling
Team (Pan-InfORM): Seyed Moghadas PhD, Institute for
Biodiagnostics, National Research Council Canada, Winnipeg, Man.;
Troy Day PhD, Department of Mathematics and Statistics, Queen’s
University, Kingston, Ont.; Chris Bauch PhD, Department of
Mathematics and Statistics, University of Guelph, Guelph, Ont.;
S. Michelle Driedger PhD, Department of Community Health Sciences,
University of Manitoba, Winnipeg, Man.; Fred Brauer PhD,
Department of Mathematics, University of British Columbia,
Vancouver, BC; Amy L. Greer PhD, Research Institute of The Hospital
for Sick Children, Toronto, Ont.; Ping Yan PhD, Public Health Agency
of Canada, Ottawa, Ont.; Jianhong Wu PhD, MITACS Centre for
Disease Modeling, York University Institute for Health Research,
Toronto, Ont.; Nick Pizzi PhD, Institute for Biodiagnostics, National
Research Council Canada, Winnipeg, Man.; and David N. Fisman MD
MPH, Ontario Agency for Health Protection and Promotion, and the
Research Institute of The Hospital for Sick Children, Toronto, Ont.
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